Revival of a Classical Topic in Differential Geometry:Envelopes of Parameterized Families of Curves andSurfaces

Th. Dana-Picard1, N. Zehavi2

1 Jerusalem College of Technology, ndp@jct.ac.il2 Weizmann Institute, nurit.zehavi@weizmann.ac.il

Classical topics in Differential Geometry like the study of 1-parameter familiesof curves and surfaces in 3D space and their envelopes have been abandoned inthe past for various reasons [4]. Nevertheless, this topic has a great interest inmathematics and in applied science. The topic has lots of applications in scienceand engineering - caustics and wave fronts, i.e. Geometrical Optics and Theoryof Singularities, robotics and kinematics, rigid motion in 2-space and in 3-space,collision avoidance, etc. Envelopes can be studied with paper-and-pencil togetherwith both a Computer Algebra System (CAS) and a Dynamical Geometry System(DGS); see recent papers in [5]. For this work, we used more than one package.

The authors performed similar work for another topic in classical DifferentialGeometry, namely isoptic curves of a given plane curve; see [1]. The influence ofthe technology was important. Here,as we will see, dynamical features are central.

Let be given a family of plane curves by an equation of the form f (x,y,c) = 0,where c is a real parameter. An envelope of the family, if it exists, is a curve tangentto every curve in the family. It can be shown that this envelope is the solution setof the system of equations {

f (x,y,c) = 0∂ f∂c f (x,y,c) = 0

(1)

Figure 1 shows the envelope of the family of lines given by the equation x+cy= c2, where c is a real parameter (it is the parabola whose equation is y=−x2/4.and the envelope of the family of circles with radius 1 and center on the parabolawhose equation is y = x2 (here the result has two components, each component isan envelope of the family of circles).

In both cases, the usage of a slider bar enables to build envelopes experimen-tally. The experimental study may enhance the understanding either of the possiblenon-existence or of the possible non-uniqueness of an envelope. An example is dis-played in Figure 2 for the family of circles whose radius is equal to 1 and whosecenter runs on an ellipse. On the left a partial construction is shown, a global con-struction appears on the right. This drawing has been obtained using the slider bar,

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(a) Lines (b) Circles

Figure 1: Exploration of an envelope in the plane

which yields a uniform spacing between circles. Another possibility offered by thesoftware is to move the center of the circle along the ellipse. In this case, spacingbetween neighboring circles is not uniform, the appearance of the envelopes beingthus slightly different.

Figure 2: Dynamical exploration of the envelope of a family of circles

The transition to parameterized families of curves and surfaces in 3D spacerely on the same techniques. For a 1-parameter family of surfaces, the definingequations for an envelope are now:{

f (x,y,z,c) = 0∂ f∂c f (x,y,z,c) = 0

(2)

New issues have to be dealt with: the general visualization problems in this case,

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the availability of appropriate features in the software, etc. For the family of sur-faces given by the equation x+ cy+ c2z = c3, where c is a real parameter, an en-velope can be found, shown in Figure 3. A cuspidal edge appears, as for every1-parameter family of planes. In order to understand the surface visually, dynami-cal features of the software are a must. Otherwise, at least two stills pictures haveto been plotted. In particular in such a case, the structure of the envelope as aruled surface can be studied using technology. This topic is sometimes not easyfor beginning students. We could check the influence of the choice of the meshfor plotting surfaces in 3D space on the students’ understanding. For example, inFigure 3, lines can be seen who are tangent to the cuspidal edge; this is a centralfeature of such an envelope.

Figure 3: Exploration of the envelope of a family of planes

Such a study provides an opportunity to discover new topics beyond the scopeof the regular curriculum, sometimes together with applications to practical situ-ations. New computation skills with technology may be developed, in particularfor the experimental aspect of the work (e.g., exploring the existence of cusps, asin Figure 1b). For this, the availability in the software of a slider bar is a centralissue. Moreover, ability to switch between different registers of representation maybe improved, within mathematics themselves (parametric vs implicit) and with thecomputer (algebraic, graphical, etc.).

An envelope may not exist (e.g. for a family of lines where the coefficients ofthe equations are affine functions of the parameter). These issues have been ob-served by the authors in sessions for in-service teachers at the Weizmann Instituteof Science.

The algebraic engine we used in different CAS was the commands based oncomputations of Gröbner bases in order 1) to solve the given system of equations,which yields a parametric representation of the envelopes, and 2) to look for an im-

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plicitization of this parametric representation. The Gröbner bases algorithms havebeen widely used by Pech [2] for other geometric problems. When such an implic-itization is not to be found, algorithms exist for an approximate implicitization (see[3]).

References[1] Dana-Picard, Th., Mann, G. and Zehavi, N. Bisoptic curves of a hyperbola, International

Journal of Mathematical Education in Science and Technology 45 (5), pp. 762-781 (2014).[2] Pech, P. Selected Topics In Geometry With Classical Vs. Computer Proving, World Scientific

(2007).[3] Pottman, H. and Peternell, M. Envelopes Computational Theory and Applications, in Pro-

ceedings of Spring Conference in Computer Graphic, Budmerice, Slovakia, pp. 3-23 (2000).[4] Thom, R. Sur la théorie des enveloppes. Journal de Mathématiques Pures et Appliquées, XLI

(2), pp. 177-192 (1962).[5] DNL 97. Derive Newsletter 97, April 2015. Available:

http://www.austromath.at/dug/dnl97.pdf

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